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Some Topologies Connected with Lebesgue Measure Séminaire De Probabilités (Strasbourg), Tome 5 (1971), P SÉMINAIRE DE PROBABILITÉS (STRASBOURG) JOHN B. WALSH Some topologies connected with Lebesgue measure Séminaire de probabilités (Strasbourg), tome 5 (1971), p. 290-310 <http://www.numdam.org/item?id=SPS_1971__5__290_0> © Springer-Verlag, Berlin Heidelberg New York, 1971, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://portail. mathdoc.fr/SemProba/) implique l’accord avec les conditions générales d’utili- sation (http://www.numdam.org/conditions). Toute utilisation commerciale ou im- pression systématique est constitutive d’une infraction pénale. Toute copie ou im- pression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ SOME TOPOLOGIES CONNECTED WITH LEBESGUE MEASURE J. B. Walsh One of the nettles flourishing in the nether regions of the field of continuous parameter processes is the quantity lim sup X. s-~ t When one most wants to use it, he can’t show it is measurable. A theorem of Doob asserts that one can always modify the process slightly so that it becomes in which case lim sup is the same separable, X~ s-t as its less relative lim where D is a countable prickly sup X, set. If, as sometimes happens, one is not free to change the process, the usual procedure is to use the countable limit anyway and hope that the process is continuous. Chung and Walsh [3] used the idea of an essential limit-that is a limit ignoring sets of Lebesgue measure and found that it enjoyed the pleasant measurability and separability properties of the countable limit in addition to being translation invariant. Doob has recently generalized and improved this, showing that there is a large class of topologies on the line enjoying similar properties, called T-topologies.(T for Lebesgue, of course:) The first three sections of the present article are an exposition of Doob’s work, including some remarks on progressive measurability,which wasn’t treated in the original. The last sections give some applications, first to potential theory and then to Markov chains. The two main properties of these topologies which are established here are first, that one can define separability relative to such a turns topology in a very natural way, and then every measurable process out to be separable, without benefit of standard modifications, and = lim secondly, if X is a measurable process, then so is Y, where sup ’ Yt Xs’ s--~ t and in fact y usually has better measurability properties than X. 291 Let T be a topology on the line R. If A C R, then AT will denote the set of T-accumulation points of A. Lebesgue measure and Lebesgue outer measure will be denoted by m and m* respectively. We will be concerned with topologies T on R satisfying (a) T is finer than the Euclidean topology; (b) a set of Lebesgue measure zero has no accumulation points; (c) for each A C R, m(A - AT) = 0 .. Such a topology will be called a T-topology. If not otherwise stated, topological notions will refer to the usual topology on the extended line. A for instance will mean the ordinary closure of A. For a real valued funct ion f on R we will def ine the T-cluster set of f at a point t to be , where N ranges over all deleted neighborhoods of t; we will denote this set by CT(f;t) .. We define also fT(t) = f(s) = max and fT(t) = f(8) = min Then we have fT a.e. (m); for fT everywhere by, and each the have definition, for b ~ 0 ,, set {t: f ( t ) >~ fT( t )+b} can no accumulation points, and hence is of zero Lebesgue measure by (c). Thus fT a.e. and similarly a.e. There are many T-topologies. One of the most useful is the essential topology L. An L-neighborhood of t is just an ordinary neighborhood minus a set of Lebesgue measure zero. A point t is an accumulation point for a set A in this topology iff 0 for each neighborhood N of t. One can easily verify that A is = = (i) L-open iff A 0 - Q ,, where 0 is open and m(Q) 0 ;; ’ (ii) L is the coarsest T-topology; (iii) the L-Borel field is exactly the Lebesgue measurable sets. Here, (iii) is a direct consequence of (i). The essential topology has some interesting connections with the ordinary topology: (iv) if A C R then AL is closed and perfect in the ordinary topology. 292 To see this, let t be in the closure of AL. If N is a neighborhood of t there exists N is also an L-neighborhood of s so . Thus tE AL. Further, by (c), so in particular N contains points of A other than t. It can be shown that this property characterizes L among T-topologies. A direct consequence of (iv) is that for a function f (v) f is upper semicontinuous in the ordinary topology; (vi) f is continuous iff f is L-continuous. Here, (v) follows by writing {fL a} as the complement of which is closed If f is it is ~ , by (iv). continuous, and if f is then f = fL = clearly L-continuous, L-continuous, f ,, so f is both.lower and upper semicontinuous, hence continuous. Two other examples of T-topologies are afforded by the right and left L-topologies,respectively denoted Lr and Neighborhoods of t are of the form N~1 and respectively, where N is an L-neighborhood of t. Results quite similar to the above hold for these topologies. Quite a different example is given by the topology Td: a point t is a Td-limit point of a set A if lim 1/2b 0. It is easily verified that Td is in a fact T-topology. T. is called the approximate topology, and it is classical that a Lebesgue measurable function is approximately continuous almost everywhere. 2.. MEASURABLE PROCESSES Let (Q,F,P) be a probability space. Let B denote the Borel sets and consider the product space RXQ. We will denote the completion of BXF with respect to mXP by’ BXF. A function X on R 03A9 to R is Borel measurable if it is BXF measurable, and is Lebesgue measurable if it is BXF measurable. An additional property shared by some but not all of the T-topologies is 293 (d) If (t,w) is Borel measurable, then so is (t,w) -~X~(t,u)) .. Here as = usual T-lim sup X(s,w).. Note that (d) is not s2014~t satisfied by the ordinary topology ! We say that two functions X and Y on RXQ are if a.e.. = indistinguishable for (P) w, X(t,w) Y(t,w), all t.. PROPOSITION 2.1 Let T be a T-topology which also satisfies (d) and let X be a Lebesgue measurable process. Then X T is indistinguishable from a Borel measurable process. PROOF. There exists a Borel measurable Y such that Y = X mXP a.e. Fubini’s a.e. w By theorem, for m{t: X(t,o)) ~ Y(t,cu)}= 0. By (b), then, for all such = for all t. By (d) Y T is Borel measurable, qed PROPOSITION 2.2 The topologies Land T along with their right and left hand topologies satisfy (d). PROOF. Suppose first that X is the indicator function of some set in BXF. Notice that in that case XL(t,03C9) = lim and = lim flim sup f-a -~ These are both B F measurable by Fubini’s theorem. For general measurable = BXF X, notice that for any real number band T L or T = T :: which is I{XLb} = lim (I{Xb-1/n}), BXF measurable, qed. The proof for the respective right and left hand topologies is entirely similar and is left to the reader. qed What one often needs in applications is progressive measurability rather than just measurability. Let be an (Ft)t~Rf increasing set of Borel sub fields of and let be the F t) i ass 0 f Bo reI sets of (-~,a). Then a function X on R 03A9 is said to be progressive measurable if for each a~R the restriction X~ of X to (here it is important to take the open interval) is measurable 294 and is progressively Lebesgue measurable if Xa is B XF measurable, the completion being again with respect to m X P .. (We are ignoring for the moment the question of whether or not X is adapted to i.e. whether or not X(t,.) is F. measurable for each fixed t. LEMMA 2.3 If X is progressively Lebesgue measurable there exists a progressively Borel measurable process X such that X = X mXP-a.e. PROOF For each integer n ~1 and each integer k there exists a function Y on 03A9 which is Bk/2n Fk/2n measurable and is mXP-a.e. to X ~ . Define Y on RXQ equal n by = on , and finally def ine X lim inf . Yn~ = Y~’~ Ck-1 ~2n, k~2n ~ X ~ , Y~ Now X plainly equals X mXP-a.e. To see that X is progressively = n and Borel measurable, note that if a k~2n ,, then for each m ~ is measurable. Thus the same is true of j f: k,, Y. B XF 1m and hence of X. For a general a, just apply this remark to a sequence of dyadic rationals which increase to a. qed THEOREM 2.4 Let T be a T-topology which satisfies (d).
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